Theorem rzal 3338

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rzal	⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		ne0i 3257	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	necon2bi 2300	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
3	2	pm2.21d 581	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
4	3	ralrimiv 2433	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∅c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by:  ralf0  3344